A small sphere of radius a is cut from a homogeneous sphere of radius R as shown in the figure. Find the position of the centre of mass of the remaining part with respect to the centre of mass of the original sphere.

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Solution

Let mass per unit volume = $σ$
Mass of bigger sphere= $σ (\frac{4}{3} π R^{3})$ at O.
Mass of smaller sphere= $σ (\frac{4}{3} π a^{3})$ at center
$X_{c o m}$ = $\frac{σ (\frac{4}{3} π R^{3}) \times 0 - σ (\frac{4}{3} π a^{3}) b}{σ (\frac{4}{3} π R^{3}) - σ (\frac{4}{3} π a^{3})}$
$= \frac{- a^{3} b}{R^{3} - a^{3}}$ directed back away from O to center of small sphere.

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A small sphere of radius a is cut from a homogeneous sphere of radius R as shown in the figure. Find the position of the centre of mass of the remaining part with respect to the centre of mass of the original sphere.

Let mass per unit volume =σMass of bigger sphere=σ(43πR3) at O.Mass of smaller sphere=σ(43πa3) at centerXcom=σ(43πR3)×0−σ(43πa3)bσ(43πR3)−σ(43πa3)=−a3bR3−a3 directed back away from O to center of small sphere.